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Stochastic calculus with respect to free Brownian motion and analysis on Wigner space
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  • Published: November 1998

Stochastic calculus with respect to free Brownian motion and analysis on Wigner space

  • Philippe Biane1 &
  • Roland Speicher2 

Probability Theory and Related Fields volume 112, pages 373–409 (1998)Cite this article

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Abstract.

We define stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of Itô's predictable representation theorem, as well as product form and functional form of Itô's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space.

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Authors and Affiliations

  1. CNRS, DMI, Ecole Normale Supérieure, 45, rue d'Ulm, 75005 Paris, France. e-mail: biane@dmi.ens,fr, , , , , , FR

    Philippe Biane

  2. Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany. e-mail: roland.Speicher@urz.uni-heidelberg.de, , , , , , DE

    Roland Speicher

Authors
  1. Philippe Biane
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  2. Roland Speicher
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Received: 6 February 1998

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Biane, P., Speicher, R. Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab Theory Relat Fields 112, 373–409 (1998). https://doi.org/10.1007/s004400050194

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  • Issue Date: November 1998

  • DOI: https://doi.org/10.1007/s004400050194

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  • Mathematics Subject Classification (1991): 60H05
  • 46L50
  • 81S25
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